DG polynomial algebras and their homological properties

In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra $mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra $mathcal{A}^{#}$ is a polynomial algebra $mathbb{k}[x_1,x_2,cdots, x_n]$ with $|x_i|=1$, for any $iin {1,2,cdots, n}$. We describe all possible differential structures on DG polynomial algebras; compute their DG automorphism groups; study their isomorphism problems; and show that they are all homologically smooth and Gorestein DG algebras. Furthermore, it is proved that the DG polynomial algebra $mathcal{A}$ is a Calabi-Yau DG algebra when its differential $partial_{mathcal{A}} eq 0$ and the trivial DG polynomial algebra $(mathcal{A}, 0)$ is Calabi-Yau if and only if $n$ is an odd integer.